Self-dual Representations Research Articles

An Euler system for GU(2,\xa01)

We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of {{,mathrm{GL},}}_3 over imaginary quadratic fields, using the cohomology of Shimura varieties for {text {GU}}(2, 1).

Irreducibility of limits of Galois representations of Saito\u2013Kurokawa type

We prove (under certain assumptions) the irreducibility of the limit sigma _2 of a sequence of irreducible essentially self-dual Galois representations sigma _k: G_{{mathbf {Q}}} rightarrow {{,mathrm{GL},}}_4(overline{{mathbf {Q}}}_p) (as k approaches 2 in a p-adic sense) which mod p reduce (after semi-simplifying) to 1 oplus rho oplus chi with rho irreducible, two-dimensional of determinant chi , where chi is the mod p cyclotomic character. More precisely, we assume that sigma _k are crystalline (with a particular choice of weights) and Siegel-ordinary at p. Such representations arise in the study of p-adic families of Siegel modular forms and properties of their limits as krightarrow 2 appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by p-adic L-values of an elliptic modular form (giving rise to rho ) which we assume are non-zero.

Multiplicities for tensor products on special linear versus classical groups

There is a natural bijective correspondence between irreducible (algebraic) selfdual representations of the special linear group with those of classical groups. In this paper, using computations done through the LiE software, we compare tensor product of irreducible selfdual representations of the special linear group with those of classical groups to formulate some conjectures relating the two. More precisely, under the natural correspondence of irreducible finite dimensional selfdual representations of $$\mathrm{SL}_{2n}({\mathbb {C}})$$ with those of $$\mathrm{Spin}_{2n+1}({\mathbb {C}})$$ , it is easy to see that if the tensor product of three irreducible representations of $$\mathrm{Spin}_{2n+1}({\mathbb {C}})$$ contains the trivial representation, then so does the tensor product of the corresponding representations of $$\mathrm{SL}_{2n}({\mathbb {C}})$$ . The paper formulates a conjecture in the reverse direction for the pairs $$(\mathrm{SL}_{2n}({\mathbb {C}}), \mathrm{Spin}_{2n+1}({\mathbb {C}})), (\mathrm{SL}_{2n+1}({\mathbb {C}}), \mathrm{Sp}_{2n}({\mathbb {C}})),$$ and $$ (\mathrm{Spin}_{2n+2}({\mathbb {C}}), \mathrm{Sp}_{2n}({\mathbb {C}})) $$ .

Faltings-Serre method on three dimensional selfdual representations

We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to $3$-dimensional Galois representations with the ground field not equal to $\mathbb{Q}$. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.

Representations of Cohomological Hall Algebras and Donaldson–Thomas Theory with Classical Structure Groups

We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $$\sigma $$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson–Thomas theory of quivers in which the quiver representations have orthogonal or symplectic structure groups. The associated invariants are called orientifold Donaldson–Thomas invariants. We prove the orientifold analogue of the integrality conjecture for $$\sigma $$ -symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of orientifold Donaldson–Thomas invariants and the freeness of the CoHM of a $$\sigma $$ -symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type $${\widetilde{A}}_1$$ . We also verify the geometric conjecture in a number of examples. Finally, we construct explicit Poincare–Birkhoff–Witt-type bases of the CoHM of finite type quivers.

Self-dual representations of SL(2,F): an approach using the Iwahori–Hecke algebra

Let F be a non-Archimedean local field and Let be an irreducible smooth Iwahori-spherical representation of G. It is easy to see that such representations are always self-dual. The space V of π admits a non-degenerate G-invariant bilinear form which is unique up to scaling. It can be shown that the form is either symmetric or skew-symmetric and we set accordingly. In this article, we use the Bernstein–Lusztig presentation of the Iwahori–Hecke algebra of G and show that

A note on self-dual representations of Sp(4,F)

Let F be a non-Archimedean local field of characteristic 0 and G=Sp(4,F). Let (π,W) be an irreducible smooth self-dual representation of G. The space W of π admits a non-degenerate G-invariant bilinear form (,) which is unique up to scaling. It can be shown that (,) is either symmetric or skew-symmetric and we set ε(π)=±1 accordingly. In this paper, we show that ε(π)=1 when π is an Iwahori-spherical representation of G.

The geometrization of N-manifolds of degree 2

This paper describes an equivalence of the canonical category of $\mathbb N$-manifolds of degree $2$ with a category of involutive double vector bundles. More precisely, we show how involutive double vector bundles are in duality with double vector bundles endowed with a linear metric. We describe then how special sections of the metric double vector bundle that is dual to a given involutive double vector bundle are the generators of a graded manifold of degree $2$ over the double base. We discuss how split Poisson $\mathbb N$-manifolds of degree $2$ are equivalent to \emph{self-dual representations up to homotopy} and so, following Gracia-Saz and Mehta, to linear splittings of a certain class of VB-algebroids. In other words, the equivalence of categories above induces an equivalence between so called \emph{Poisson involutive double vector bundles}, which are the dual objects to metric VB-algebroids, and Poisson $\mathbb N$-manifolds of degree $2$.

Twisted eigenvarieties and self-dual representations

Twisted eigenvarieties and self-dual representations

Descent Construction for GSpin groups

In this paper we provide an extension of the theory of descent of Ginzburg-Rallis- Soudry to the context of essentially self-dual representations, that is representations which are isomorphic to the twist of their own contragredient by some Hecke character. Our theory supplements the recent work of Asgari-Shahidi on the functorial lift from (split and quasisplit forms of) GSpin(2n) to GL(2n).

Self-dual representations of $\operatorname {SL}(n,F)$

Self-dual representations of $\operatorname {SL}(n,F)$

Self-dual quiver moduli and orientifold Donaldson-Thomas invariants

Motivated by the counting of BPS states in string theory with orientifolds, we study moduli spaces of self-dual representations of a quiver with contravariant involution. We develop Hall module techniques to compute the number of points over finite fields of moduli stacks of semistable self-dual representations. Wall-crossing formulas relating these counts for different choices of stability parameters recover the wall-crossing of orientifold BPS/Donaldson-Thomas invariants predicted in the physics literature. In finite type examples the wall-crossing formulas can be reformulated in terms of identities for quantum dilogarithms acting in representations of quantum tori.

Diagram automorphisms of quiver varieties

We show that the fixed-point subvariety of a Nakajima quiver variety under a diagram automorphism is a disconnected union of quiver varieties for the ‘split-quotient quiver’ introduced by Reiten and Riedtmann. As a special case, quiver varieties of type D arise as the connected components of fixed-point subvarieties of diagram involutions of quiver varieties of type A. In the case where the quiver varieties of type A correspond to small self-dual representations, we show that the diagram involutions coincide with classical involutions of two-row Slodowy varieties. It follows that certain quiver varieties of type D are isomorphic to Slodowy varieties for orthogonal or symplectic Lie algebras.

Fock parafermions and self-dual representations of the braid group

We introduce and describe in second quantization a family of particle species with \(p=2,3,\dots\) exclusion and \(\theta=2\pi/p\) exchange statistics. We call these anyons Fock parafermions, because they are the particles naturally associated to the parafermionic zero-energy modes, potentially realizable in mesoscopic arrays of fractional topological insulators. Their second-quantization description entails the concept of Fock algebra, i.e., a Fock space endowed with a statistical multiplication that captures and logically correlates these anyons' exclusion and exchange statistics. As a consequence normal-ordering continues to be a well-defined operation. Because of its relevance to topological quantum information processing, we also derive families of self-dual representations of the braid group for any $p$, with the Gaussian representation being a special case. The self-dual representations can be realized in terms of local quadratic combinations of either parafermions or Fock parafermions, an important requisite for physical implementation of quantum logic gates.

Self-dual representations with vectors fixed under an Iwahori subgroup

Let G be the group of F-points of a split connected reductive F-group over a non-Archimedean local field F of characteristic 0. Let π be an irreducible smooth self-dual representation of G. The space W of π carries a non-degenerate G-invariant bilinear form (,) which is unique up to scaling. The form is easily seen to be symmetric or skew-symmetric and we set ε(π)=±1 accordingly. In this article, we show that ε(π)=1 when π is a generic representation of G with non-zero vectors fixed under an Iwahori subgroup I.

Self-dual representations of division algebras and Weil groups: A contrast

Irreducible selfdual representations of any group fall into two classes: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations $\sigma$ of the Weil group of a local field $k$ of dimension $n$ with the irreducible representations $\pi$ of the invertible elements of a division algebra $D$ over $k$ of index $n$, takes selfdual representations to selfdual representations. In this paper we use global methods to study how the Langlands correspondence behaves relative to this distinction among selfdual representations. We prove in particular that for $n$ even, $\sigma$ is symplectic if and only if $\pi$ is orthogonal. More generally, we treat the case of ${\rm GL}_m(B)$, for $B$ a division algebra over $k$ of index $r$, and $n=mr$.

On deformation rings of residually reducible Galois representations and R = T theorems

We introduce a new method of proof for R = T theorems in the residually reducible case. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ 0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ 1 and ρ 2. Under some assumptions on Selmer groups associated with ρ 1 and ρ 2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaïche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R = T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over Q.

Self-dual representations of some dyadic groups

Let F be a non-Archimedean local field of residual characteristic two and let d be an odd positive integer. Let D be a central F-division algebra of dimension d2. Let π be one of: an irreducible smooth representation of D × , an irreducible cuspidal representation of GLd(F), an irreducible smooth representation of the Weil group of F of dimension d. We show that, in all these cases, if π is self-contragredient then it is defined over \({\mathbb Q}\) and is orthogonal. We also show that such representations exist.

Corps de nombres peu ramifiés et formes automorphes autoduales

Let S S be a finite set of primes, p p in S S , and Q S \mathbb {Q}_S a maximal algebraic extension of Q \mathbb {Q} unramified outside S S and ∞ \infty . Assume that | S | ≥ 2 |S|\geq 2 . We show that the natural maps \[ Gal ⁡ ( Q p ¯ / Q p ) → Gal ⁡ ( Q S / Q ) \operatorname {Gal}(\overline {\mathbb {Q}_p}/\mathbb {Q}_p) \rightarrow \operatorname {Gal}(\mathbb {Q}_S/\mathbb {Q}) \] are injective. Much of the paper is devoted to the problem of constructing self-dual automorphic cuspidal representations of GL ⁡ ( 2 n , A Q ) \operatorname {GL}(2n,\mathbb {A}_{\mathbb {Q}}) with prescribed properties at all places, which we study via Arthur’s twisted trace formula. The techniques we develop also shed some light on the orthogonal/symplectic alternative for self-dual representations of GL ⁡ ( 2 n ) \operatorname {GL}(2n) .

The classification of reversible-equivariant steady-state bifurcations on self-dual spaces

AbstractIn this paper we apply singularity theory methods to the classification of reversible-equivariant steady-state bifurcations depending on one real parameter. We assume that the group of symmetries and reversing symmetries is a compact Lie group Γ, and the equivalence is defined in order to preserve these symmetries and reversing symmetries in the normal forms and their unfoldings. When the representation of Γ is self-dual, we show that the classification can be reduced to the standard equivariant context. In this case, we establish a one-to-one association between the classification of bifurcations in the reversible-equivariant context and the classification of purely equivariant bifurcations related to them. As an application of the results, we obtain the classification of self-dual representations ofZ2⊕Z2andD4on the plane.